Non- Parametric Test
A Non-Parametric tests is a one of the part of Statistical tests that non-parametric test does not assume any particular distribution for analyzing the variable. unlike the parametric test are based on the assumption like normality or other specific distribution of the variable. Non-parametric test is based on the rank, order, signs, or other non-numerical data. we know both test parametric and non-parametric, but when use particular test? answer is that if the assumption of parametric test are violated such as data is not normally distributed or sample size is small. then we use Non-parametric test they can used to analyse categorical data or ordinal data and data are obtained form in field like psychology, sociology and biology. For the analysis use the some non-parametric test that are Wilcoxon signed-ranked test, mann-whiteny U test, sign test, Run test, Kruskal-wallis test. but the non-parametric test have lower statistical power than parametric test when assumption parametric test valid.
First we see the assumption of parametric and non-parametric test assumptions.
Assumption of Parametric test:-
following are the key assumption of Parametric test:
1. Normality: The data must be normally distributed. mean that data must be follow normal distribution. and the data should be bell-shaped, with the majority of the data points falling near to the mean.
2. Independence: the data is should be independent, mean the data point are should be unrelated to each other. and the value of one variable is not affected by the value of other variable.
3. Homoscedasticity: that mean homogeneity of variance, the variance of the data should be equal across group or sample being compared. this means that the spread of the data should be similar in different groups.
4. Continuous data: parametric tests are designed for continuous data, that means that the should me measured in scale that has equal interval between each value.
when these assumption are meet then parametric test is very powerful and accurate. however the assumption are violated any one assumption it gives inaccurate results and misleading conclusions. in that case we use the Non-parametric Test.
Non-Parametric Test:
Non-parametric test are statistical test that dose not required any specific assumption about the probability distribution of variable being analyzed. however the non-parametric test have some assumptions as:
1. Independence: the data is should be independent, mean the data point are should be unrelated to each other. and the value of one variable is not affected by the value of other variable.
2.Random Sampling: the data should be obtained through random sampling. that means the each observation have equal chance of being selected, and sampling should be representative of the population being selected.
3.Ordinal Data: Non-parametric test is designed for ordinal data. that mean if data is ranked or measured in ordinal scale then we test.
4.Homogeneity of variance: the variance of the group being selected for comparison it should be similar.
Non-Parametric test is more flexible and robust than parametric test. because they do not based on strict assumption about distribution of the data. for accurate result carefully consider assumptions.
Types of Non-Parametric Test :
the non-parametric test is Broadly divided into three categories as:
1. One-sample Test
2.Two-sample test
3. K-sample test
we see all these types of Non-Parametric test.
ONE-SAMPLE TESTS
RUN TEST
One of the fundamental assumptions of the parametric test is that the observed data are random and test statistic and the subsequent analysis are based on this assumption. It is always better to check whether this assumption is true or not.
A very simple tool for checking this assumption is run test. This section is devoted to throw light on the run test. Before discussing the run test first we have to explain what we mean by a “run”.
A run in observations, is defined as a sequence of letters or symbols of one kind, immediately preceded and succeeded by letters of other kind or no letters. For example a sequence of two letters H and T as given below:
HHTHTTTHTHHHTTT
In this sequence, we start with first letter H and go up to other kind of letter, that is, T. In this way, we get first run of two H’s. Then we start with this T and go up to other kind of letter, that is, H. Then we get a run of one T and so on and finally a run of three T’s. In all, we see that there are eight runs. And it is denoted as r=8. it is shown as below.
HH T H TTT H T HHH TTT 1 2 3 4 5 6 7 8
Under the run test we Judged randomness of observations by using number of runs in the observed sequence. Too few runs indicates that there is some clustering or trend and too large runs indicates that there is some kind of repeated or cycles according to some patterns.
for example, the following sequence of H's and T's is obtained when tossing a coin 10 times.
HHHHHHTTTT 1 2
in the above sequence we see there is only 2 run's of 6 heads and 4 Tails, hence from this we say the similar item tend to cluster together, therefore such sequence of observation is not considered as random. now the anther sequence of 10 tosses is as following:
H T H T H T H T H T 1 2 3 4 5 6 7 8 9 10 .
this numbers indicates the runs for all observation in this sequence there are 10runs of 5 runs of one head each and 5 runs of one tail each. this sequence is cloud not be considered as random because there are too many runs they indicates the pattern.
T HHH TT H T HH T HHH TT H T H T 1 2 3 4 5 6 7 8 9 10 111213
Here neither the number of runs too small nor too large, this type of sequence may be considered as random.
Assumptions:
Run test make the following assumptions:
(i) Observed data should be such that we can categorise the observations into two mutually exclusive types.
(ii) The variable under study is continuous.
Procedure for RUN test
Let X1,X2,...,Xn be a set of no observations arranged in the order in which they occur. Generally, we are interested to test whether a population or a sample or a sequence is random or not. So here we consider only two-tailed case. Thus, we can take the null and alternative hypotheses as
H0 : The observations are random
H1 : The observations are not random [two-tailed test]
test consist following steps:
Step 1: First of all, we check the form of the given dada that the given data are in symbolical form such as sequence of H and T, A and B, etc. or in the numeric form. If the data in symbolical form then it is ok, but if data in numeric form then first we convert numeric data in symbolical form. For this, we calculate median of the given observations by using either of the following formula given below
Median = size of [(n+1) /2 ] th observation.
provided observations should be either in ascending or descending order of magnitude.
After that, we replace the observations which are above the median by a symbol ‘A’ (say) and the observation which are below the median by a symbol ‘B’ (say) without altering the observed order. The observations which are equal to median are discarded form the analysis and let reduced size of the sample denoted by n.
Step 2: Counts number of times first symbol (A) occurs and denote it by n1
Step 3: Counts number of times second symbol (B) occurs and denote it by n2 where, n = n1+n2
Step 4: For testing the null hypothesis, the test statistic is the total number of runs so in this step we count total number of runs in the sequence of symbols and denote it by R.
Step 5: Obtain critical values of test statistic corresponding n1,n2 at α % level of significance under the condition that null hypothesis is true. From the Table of critical value for run test is used to obtain respectively lower (RL) and upper (RU) critical values of the number of runs for a given combination of n1 and n2 at 5% level of significance.
Note1: Generally, critical values for run test are available at 5% level of
significance so we test our hypotheses for 5% level of significance.
Step 6: Decision Rule:
To take the decision about null hypothesis, the test statistic is compared with the critical (tabulated) values.
If the observed number of runs(R) is either less than or equal to the lower critical value (RL) or greater than or equal to the upper critical value (RU), that is, if R < RL or R > RU then we reject the null hypothesis at 5% level of significance.
If R lies between Rl and Ru , that is, Rl < R> Ru, then we Accept
null hypothesis at 5% level of significance
Non- Parametric Test
2. Paired Sign Test: Or two sample sign test
In social sciences the two related groups are paired and we interested to examine the difference between two related groups. if the observations are recorded or data are available as after and before type means the data are recorded before diet and after diet that time we get paired observations of same variable or item, in this situations we use Paired T- Test. if the Assumption of the T-Test are fulfil, if Assumption of T-Test is not fulfil that time we use the paired sign test, that mean the paired sing test is alternative Non-Parametric test for Parametric Paired T-Test. in other word in some situation paired T-Test is not applicable then we use the Non- parametric Paired sign test. e.g. let's say you want to know if a new exercise program is effective in reducing body fat, you randomly select 10 participants and measure there body fat, then you have them follow the exercise program for 4 weeks and measure there body fat. this give before and after data in that situation we use paired sign test if data does not fulfil the assumption of Paired T-Test. also sign test is applicable when data are ordinal or given in symbolically.(i.e. + or -, a or b)
Assumptions:
If the data follow the following assumption then we use Paired Sign Test:
1. the pair of observations are independents,
2. the measurement of variable is at ordinal scale. and variable under study is continuous.
let (X1, Y1),(X2, Y2), .......(Xn, Yn), random sample of size n independent and continuous items or units, each observation is measured in before and after the (e.g. after and before training, diet, or treatment etc. here we want to test the their is effect of diet, training etc. for that we can take the null and alternative hypothesis.
H0 :μ1=μ2 VS
H0 :μ1≠ μ2
this is two tailed test
the Paired Sign Test has same procedure as Sign test. they are divided into following steps.
Step I: the sign test is based on Sign hence firstly we converting the data into sequence of plus and Minus signs. for this we compare the observations Xi and Yi if the observation Xi > Yi then we take plus (+) sign and if the observation Xi < Yi then we take minus (-) sign. and one of the observation Xi is equal to Yi or Xi = Yi then we removed that observation form the data then the sample size is reduced is denoted as n.
Step II: In this step we count the number of (+) plus and number of (-) minus signs. and it is denoted as S+ And S- , S+ for number of plus signs and S- for number of minus signs.
Step III: Now we consider the null hypothesis is true then on the basis of postulated value of median we expect that the value of variable is greater than median mean we get plus sign then the number of plus sign it consider as success and number of minus sign is consider as failure approximately equal. then the distribution of sign is binomial distribution with parameter (n, p=0.5). for simplicity we consider smaller number of sign. that means the number of plus sign is less than number of minus sign then plus sign is success and minus is failure. similarly if minus signs are less than number of plus sign then minus sign is success and number of plus sign is failure.
Step IV:
i. Small Sample Test : (i.e. n is less than or equal to 20 ).
if the number of observations are is less than or equal to 20 (i.e. n £ 20 ). is called small sample test.
for Decision about hypothesis we we the p-value and it is determined as
P-value = 2 P(S £ s) and where s is equal to S = min ( S+,S-) this is foe two tailed test.
(note that for the two sample test we consider P-value = 2P(S £ s), and for any one tailed test the test Statistics is P-value = P(S £ s) this is small change)
if the number of observations are is less than or equal to 20 (i.e. n £ 20 ). is called small sample test.
if the p-value is less than or equal to a% level of significance then we reject the null hypothesis at a% level of significance otherwise accept the null hypothesis.
Large sample test:(i.e. n greater than 20 )
if the number of observation is greater than 20 we use large sample test (i.e. n >20 ).
for large sample test we use normal approximation for binomial distribution.
as E(S) = n*1/2 = n/2
and S.D.( S) = n*1/4= n/4
the normal approximation is z test gives as
z = (S - E(S))/S.D.(S)
Z = (S - (n/2))/(n/4)
then we comparing the calculated and tabulated value of z. at a% level of significance.
if calculated z is less than or equal to tabulated (critical value ) then we accept null hypothesis other wise reject the null hypothesis.
In the next part we discuss the examples of Paired sign test.
next part coming soon
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